Holographic Formulation of 3D Metric Gravity with Finite Boundaries

Abstract

In this work we construct holographic boundary theories for linearized 3D gravity, for a general family of finite or quasi-local boundaries. These boundary theories are directly derived from the dynamics of 3D gravity by computing the effective action for a geometric boundary observable, which measures the geodesic length from a given boundary point to some centre in the bulk manifold. We identify the general form for these boundary theories and find that these are Liouville like with a coupling to the boundary Ricci scalar. This is illustrated with various examples, which each offer interesting insights into the structure of holographic boundary theories. Contents* Electronic address: sasanteATperimeterinstitute.ca † Electronic address: bdittrichATperimeterinstitute.ca ‡ Electronic address: fhopfmuellerATperimeterinstitute.ca VI. Twisted thermal AdS space with finite boundary 23 A. Equations of motion and evaluation of the action 23 B. One loop correction from the dual field 25 VII. Flat space with spherical boundary 25 A. Solutions to the equations of motion 26 B. Evaluation of the action 27 VIII. Discussion and outlook 29 A. Conventions and Gauss-Codazzi relations 31 B. Vector basis for induced perturbations 31 C. Second order of the Hamilton-Jacobi functional 34 D. Evaluation of the commutator [∂ ⊥ , ∆] 36 E. Solutions of the equations of motion for the case 2 R = 0 37 F. Equations of motion in spherical coordinates 40 G. Lagrange multiplier dependent boundary terms 42 H. Geodesic length to first order in metric perturbations 43 I. Smoothness conditions for the metric at r = 0 44 J. On effective actions 45 K. Spherical tensor harmonics 46 References 46 9The boundary fluctuations for which one can and cannot find solutions for lapse and shift can be read off from (5.13). Eg. allowing only for non-vanishing fluctuations γ θθ still allows for a solution.

Extended actions, dynamics of edge modes, and entanglement entropy

“…Recently, many new results revealing important insights into the role of edge modes have been obtained, both at finite distance and at infinity. At finite distance, there have been successful definitions of quasi-local holography through the path integral for quantum gravity [7][8][9][10][11][12][13], leading to boundary models which can be thought of as capturing the dynamics of the edge modes. There have also been efforts to characterize, for local subsystems, the most general boundary symmetry algebras spanned by the edge modes [14][15][16][17][18][19][20][21], with potentially important consequences for quantum gravity [22,23].…”

Quantum geometry from higher gauge theory

“…In particular, the 2-flatness constraint for the 2-connection takes in the discrete a very simple form (a sum over momenta 5 ) and, through its action on the tetrad field, it can be readily interpreted as a combination of the diffeomorphism and Hamiltonian constraint for its action on the tetrad field. 6 More precisely, the 2-flatness constraint generates spacetime translations for the quantum-flat simplicial geometry -thus generalizing for the first time to four dimensions what is understood as the action of the diffeomorphism and Hamiltonian constraints in three dimensions [24,33,72].…”

Extended actions, dynamics of edge modes, and entanglement entropy

“…Recently, many new results revealing important insights into the role of edge modes have been obtained, both at finite distance and at infinity. At finite distance, there have been successful definitions of quasi-local holography through the path integral for quantum gravity [7][8][9][10][11][12], leading to boundary models which can be thought of as capturing the dynamics of the edge modes. There have also been efforts to characterize, for local subsystems, the most general boundary symmetry algebras spanned by the edge modes [13][14][15][16][17][18][19][20], with potentially important consequences for quantum gravity [21,22].…”